A set with binary composition denoted multiplicatively is a group if
(i) The composition is associative.
(ii) For every pair of elements , the equations and have unique solutions in .
Proof: Binary operation implies that the set , under consideration is closed under the operation. Now to prove that the set is a group we have to show the left identity exists and each element of possesses left inverse with respect to the operation under consideration.
It is given that for every pair of elements the equation has a solution in . Therefore, say such that .
Now, let us suppose that is any arbitrary element of . Therefore, there exist such that .
Therefore there exist such that
is the left identity.
Now, let be an element. is inverse of in .
Let such that , then as has got solution in .
Thus is the left inverse of in . Therefore each element of possesses left inverse. Hence is a group for the given composition if the postulate and (ii) are satisfied.